Including the effect of LNV from the dimension-five Weinberg operator [1]},
the low-energy effective Lagrangian at scale \(\mu \gtrsim \Lambda _\chi \sim 1\,\text{GeV}\)  is given by
\({\cal L}_{\rm eff} = {\cal L}_{\rm QCD} \ -\left\lbrace  \ 2 \sqrt{2} G_F V_{ud} \ \bar{u}_L \gamma ^\mu d_L \, \bar{e}_L \gamma _\mu \nu _{eL}\ + \ \frac{1}{2} m_{\beta \beta }^* \ {\nu }_{eL}^T C \nu _{eL}\ - \ C_L \, O_{L} +{\rm h.c.}\right\rbrace \,.\) 
